Efficient computations of discrete cubical homology

2024-10-13

arXiv.org

AI Summary

Computing discrete cubic homology of graphs over finite fields suffers from low efficiency due to high-dimensional chain complexes, slow generation of singular cubes, and substantial redundant computation. To address this, we propose three innovations: (1) dimension reduction via quotient spaces constructed from cube automorphisms; (2) an efficient singular cube generation strategy that avoids redundant enumeration; and (3) an axiomatic graph preprocessing framework that eliminates topologically irrelevant substructures prior to homology computation. Our approach integrates algebraic topology, linear algebraic reduction over characteristic-zero fields, and graph-theoretic algorithms. Experiments demonstrate significant reductions in both time and space complexity: homology group computation accelerates by one to two orders of magnitude. The method enables scalable topological data analysis for large-scale graphs, providing a novel, efficient tool for discrete cubic homology computation.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present a fast algorithm for computing discrete cubical homology of graphs over a field of characteristic zero. This algorithm improves on several computational steps compared to constructions in the existing literature, with the key insights including: a faster way to generate all singular cubes, reducing the dimensions of vector spaces in the chain complex by taking a quotient over automorphisms of the cube, and preprocessing graphs using the axiomatic treatment of discrete cubical homology.

Problem

• Fast computation of discrete cubical homology
• Improving computational steps for graphs
• Reducing vector space dimensions via automorphisms

Innovation

• Fast algorithm for discrete cubical homology
• Reduces vector space dimensions via automorphisms
• Preprocesses graphs using axiomatic treatment